(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 10.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 1081808, 20346] NotebookOptionsPosition[ 1076872, 20183] NotebookOutlinePosition[ 1077285, 20201] CellTagsIndexPosition[ 1077242, 20198] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "Systems of first order differential equations:\n", StyleBox["Equilibrium solutions (Critical point, Steady State Solution)\n\ Types of critical points\nStability of equilibrium solutions", FontSize->24, FontWeight->"Bold", FontSlant->"Italic", FontColor->GrayLevel[0]] }], "Title", CellChangeTimes->{{3.699685254175375*^9, 3.699685266835086*^9}, 3.699685311852395*^9, {3.6996854059114637`*^9, 3.6996854088621473`*^9}, { 3.699685472561427*^9, 3.699685473764043*^9}}], Cell[CellGroupData[{ Cell["Predator-Prey System: ", "Section", CellChangeTimes->{{3.622790466683278*^9, 3.622790476625743*^9}, { 3.622790511711554*^9, 3.6227905335575237`*^9}, {3.6227906625672903`*^9, 3.622790669390012*^9}, {3.6538711094179993`*^9, 3.65387111076064*^9}, { 3.653874243652176*^9, 3.653874255935793*^9}, {3.654463985629437*^9, 3.654464002198286*^9}, {3.654464069364059*^9, 3.654464102635427*^9}, { 3.655176007208185*^9, 3.655176007686988*^9}, {3.699685321027286*^9, 3.699685336779744*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "Consider the system usually described as a predator - prey model (Voltera, \ 1931):\n ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ FractionBox["dx", "dt"], FontSize->36], StyleBox["=", FontSize->24], StyleBox[ RowBox[{ RowBox[{"0.2", "x"}], "-", RowBox[{"0.7", "xy"}]}], FontSize->24]}], TraditionalForm]], FontSize->36], StyleBox[",", FontSize->36], " ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox["dy", "dt"], StyleBox["=", FontSize->24], StyleBox[ RowBox[{ RowBox[{ RowBox[{"-", "0.4"}], "y"}], "+", RowBox[{"0.5", "xy"}]}], FontSize->24]}], TraditionalForm]], FontSize->36], "." }], "Subsubsection", CellChangeTimes->{{3.6544642408637457`*^9, 3.654464263463875*^9}, { 3.65446429674226*^9, 3.6544644794677*^9}, {3.654464601362609*^9, 3.654464602290267*^9}, {3.654464650514975*^9, 3.654464665223145*^9}}], Cell["\<\ where you can think of the function x(t) as representing a population of \ rabbits that naturally grow at a rate proportional to their population (that \ is, exponential growth in the absence of predators) and the function y(t) as \ representing a population of foxes that naturally decline (that is, \ exponential decay in the absence of prey). The term xy in both equations is \ proportional to the number of likely encounters of the two population in a \ certain environment. Encounters are detrimental to the rabbits and beneficial \ to the foxes.\ \>", "Text", CellChangeTimes->{{3.654464605624176*^9, 3.654464628508052*^9}, { 3.654465226697503*^9, 3.6544652297351027`*^9}, {3.6544652632054167`*^9, 3.654465270382663*^9}}, TextJustification->1., FontSize->16] }, Open ]], Cell["\<\ This is a nonlinear system, so an explicit solution is very hard to find. \ \>", "Subsubsection", CellChangeTimes->{{3.6996853553487387`*^9, 3.699685395390071*^9}, { 3.699686224323613*^9, 3.699686225017097*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Systems of first order linear differential equations", FontColor->GrayLevel[0]]], "Section", CellChangeTimes->{{3.622973133683782*^9, 3.622973152291214*^9}, { 3.654913941545538*^9, 3.654913957424914*^9}}], Cell[TextData[{ "Consider a system S of linear differential equations ", Cell[BoxData[ FormBox[ TagBox[GridBox[{ {"\[Piecewise]", GridBox[{ { RowBox[{ RowBox[{"x", "'"}], "=", RowBox[{"ax", "+", "by"}]}], RowBox[{"(", "1", ")"}]}, { RowBox[{ RowBox[{"y", "'"}], "=", RowBox[{"cx", "+", "dy"}]}], RowBox[{"(", "2", ")"}]} }, AllowedDimensions->{2, Automatic}, Editable->True, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.84]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}, Selectable->True]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.35]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "Piecewise", DeleteWithContents->True, Editable->False, SelectWithContents->True, Selectable->False], TraditionalForm]]] }], "Subsection", CellChangeTimes->{{3.6227922426259937`*^9, 3.62279231309624*^9}, { 3.6227927369105253`*^9, 3.622792744486435*^9}, {3.622805954711268*^9, 3.6228060525702353`*^9}, {3.622815694267383*^9, 3.6228156945063753`*^9}, { 3.622973160147009*^9, 3.622973167010988*^9}, {3.622973244164934*^9, 3.6229732511893044`*^9}, {3.654839517653059*^9, 3.654839538308358*^9}, { 3.654840308944374*^9, 3.654840346114191*^9}}], Cell[TextData[{ "The functions x and y are functions of time t and the derivatives are also \ taken with respect to the variable t. The numbers a,b,c,d are real numbers \ and they are called the coefficients, or the parameters of the system. Our \ system above can also be written as\n ", Cell[BoxData[ FormBox[ TagBox[GridBox[{ {"\[Piecewise]", GridBox[{ { RowBox[{ RowBox[{ RowBox[{"x", "'"}], RowBox[{"(", "t", ")"}]}], "=", RowBox[{ RowBox[{"ax", "(", "t", ")"}], "+", RowBox[{"by", "(", "t", ")"}]}]}], RowBox[{"(", "1", ")"}]}, { RowBox[{ RowBox[{ RowBox[{"y", "'"}], RowBox[{"(", "t", ")"}]}], "=", RowBox[{ RowBox[{"cx", "(", "t", ")"}], "+", RowBox[{"dy", "(", "t", ")"}]}]}], RowBox[{"(", "2", ")"}]} }, AllowedDimensions->{2, Automatic}, Editable->True, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.84]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}, Selectable->True]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.35]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "Piecewise", DeleteWithContents->True, Editable->False, SelectWithContents->True, Selectable->False], TraditionalForm]]], " or ", Cell[BoxData[ FormBox[ TagBox[GridBox[{ {"\[Piecewise]", GridBox[{ { RowBox[{ StyleBox[ FractionBox["dx", "dt"], FontSize->24], "=", RowBox[{ RowBox[{"ax", "(", "t", ")"}], "+", RowBox[{"by", "(", "t", ")"}]}]}], RowBox[{"(", "1", ")"}]}, { RowBox[{ StyleBox[ FractionBox["dy", "dt"], FontSize->24], "=", RowBox[{ RowBox[{"cx", "(", "t", ")"}], "+", RowBox[{"dy", "(", "t", ")"}]}]}], RowBox[{"(", "2", ")"}]} }, AllowedDimensions->{2, Automatic}, Editable->True, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.84]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}, Selectable->True]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.35]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "Piecewise", DeleteWithContents->True, Editable->False, SelectWithContents->True, Selectable->False], TraditionalForm]]], " , but the dependence on t is usually not explicitly written." }], "Subsection", CellChangeTimes->{{3.622806089809124*^9, 3.6228061016956873`*^9}, { 3.622806146675913*^9, 3.622806280122282*^9}, {3.6228063317840443`*^9, 3.622806352340406*^9}, {3.622973262660453*^9, 3.62297331713972*^9}, 3.6229751603254747`*^9, 3.622986200393735*^9, {3.654839546056003*^9, 3.654839555044265*^9}, {3.69968549150638*^9, 3.699685699056157*^9}, { 3.699685896297086*^9, 3.699685909991056*^9}}, FontColor->GrayLevel[0]], Cell[TextData[{ "Definition: ", StyleBox["The phase portrait of a systems of linear differential equations \ is a representative set of its solutions (trajectories, integral curves), \ plotted as parametric curves (with t as the parameter) on the Cartesian plane \ xy, tracing the path of each particular solution (x, y) = (x(t), y(t)), \ \[Minus]\[Infinity] < t < \[Infinity] . ", FontColor->GrayLevel[0]] }], "Subsection", CellChangeTimes->{{3.654839651603204*^9, 3.654839897851862*^9}, { 3.654841010997834*^9, 3.6548410242677393`*^9}, {3.654841069442794*^9, 3.654841083328396*^9}, {3.699685735667638*^9, 3.699685823196096*^9}}, TextJustification->1.], Cell[CellGroupData[{ Cell[TextData[{ "Phase Portrait in Mathematica: ", StyleBox["The Phase Portrait is obtained in ", FontColor->GrayLevel[0]], StyleBox["Mathematica", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" using the command StreamPlot[..].", FontColor->GrayLevel[0]] }], "Subsection", CellChangeTimes->{{3.699685828577989*^9, 3.699685872806992*^9}}], Cell[TextData[{ "Similar to a vector field (obtained in ", StyleBox["Mathematica", FontSlant->"Italic"], " using the command VectorPlot), a phase portrait is a graphical tool to \ visualize how the solutions of a given system of differential equations would \ behave in the long run. We can classify the behavior of solutions and the \ type and stability of the equilibrium solution of a given linear system by \ looking at the phase portrait and analyzing the properties of the coefficient \ matrix A = ", Cell[BoxData[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"a", "b"}, {"c", "d"} }, GridBoxDividers->{ "Columns" -> {False, {True}, False}, "ColumnsIndexed" -> {}, "Rows" -> {{False}}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}]]], ". " }], "Subsubsection", CellChangeTimes->{{3.654839949189966*^9, 3.654839982345683*^9}, { 3.654840014938785*^9, 3.654840151382045*^9}, {3.6548410611017714`*^9, 3.654841066322191*^9}, {3.6548410984238787`*^9, 3.654841099560473*^9}}, TextJustification->1.] }, Open ]], Cell[TextData[{ "Advantage: ", StyleBox["We can solve these systems explicitly!", FontColor->GrayLevel[0]] }], "Subsection", CellChangeTimes->{{3.69968612548598*^9, 3.699686191536026*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Equilibrium Solution", FontColor->GrayLevel[0]]], "Section", CellChangeTimes->{{3.654840182528174*^9, 3.6548402040876017`*^9}}], Cell[TextData[{ "Definition: ", StyleBox["An equilibrium solution of the system S is a point (x,y) where x\ \[CloseCurlyQuote]=0 and y\[CloseCurlyQuote]=0. An equilibrium solution is a \ constant solution of the system, and is sometimes also called a critical \ point, or a steady state solution. \n", FontColor->GrayLevel[0]] }], "Subsection", CellChangeTimes->{{3.654840268362956*^9, 3.6548402925296783`*^9}, { 3.654840355360712*^9, 3.6548407879219313`*^9}, {3.65484095422857*^9, 3.654840992126856*^9}, {3.654841169906949*^9, 3.654841171611444*^9}, { 3.6996860578931*^9, 3.699686062899995*^9}}, TextJustification->1.], Cell[CellGroupData[{ Cell["How to find the equilibrium solutions of a system :", "Subsection", CellChangeTimes->{{3.699686018102326*^9, 3.699686054758553*^9}}], Cell[TextData[{ StyleBox["For our linear system of differential equations, an equilibrium \ solution occurs at each solution of the system (of homogeneous algebraic \ equations) ax+by=0 and cx+dy=0. Equivalently, we need to solve the matrix \ equation A", FontColor->GrayLevel[0]], Cell[BoxData[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"x"}, {"y"} }, GridBoxDividers->{ "Columns" -> {False, {True}, False}, "ColumnsIndexed" -> {}, "Rows" -> {{False}}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}]], FontColor->GrayLevel[0]], StyleBox["=", FontColor->GrayLevel[0]], Cell[BoxData[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0"}, {"0"} }, GridBoxDividers->{ "Columns" -> {False, {True}, False}, "ColumnsIndexed" -> {}, "Rows" -> {{False}}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}]], FontColor->GrayLevel[0]], ", where A=", Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ {"a", "b"}, {"c", "d"} }], ")"}], TraditionalForm]], FormatType->"TraditionalForm"], " is called the coefficient matrix of the system. ", StyleBox["As we have seen, such a system has exactly one solution, located \ at the origin ", FontColor->GrayLevel[0]], Cell[BoxData[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0"}, {"0"} }, GridBoxDividers->{ "Columns" -> {False, {True}, False}, "ColumnsIndexed" -> {}, "Rows" -> {{False}}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}]], FontColor->GrayLevel[0]], StyleBox[", if and only if the matrix A is invertible. If A is not \ invertible, then there are infinitely many solutions. ", FontColor->GrayLevel[0]] }], "Subsubsection", CellChangeTimes->{{3.654839949189966*^9, 3.654839982345683*^9}, { 3.654840014938785*^9, 3.654840151382045*^9}, {3.6548410611017714`*^9, 3.654841066322191*^9}, {3.6548410984238787`*^9, 3.654841099560473*^9}, { 3.654841143848014*^9, 3.65484115838113*^9}, {3.654841253239213*^9, 3.6548413142693357`*^9}, 3.699685992738379*^9, {3.699686331439595*^9, 3.699686371390624*^9}}, TextJustification->1.], Cell[TextData[{ "\nRecall also that the following statements are equivalent: \n", StyleBox["A is invertible", FontColor->GrayLevel[0]], " \[DoubleLeftRightArrow] ", StyleBox["the determinant of the matrix det (A) is different from 0", FontColor->GrayLevel[0]], " \[DoubleLeftRightArrow] ", StyleBox["0 is an eigenvalue of A", FontColor->GrayLevel[0]], ". " }], "Subsubsection", CellChangeTimes->{{3.6548413234968777`*^9, 3.6548413350153303`*^9}, { 3.654841457888879*^9, 3.654841550509378*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Classification and Stability of Critical Points", FontColor->GrayLevel[0]]], "Section", CellChangeTimes->{{3.654840897086129*^9, 3.654840932177696*^9}}], Cell[TextData[StyleBox["Assume that det(A) \[NotEqual] 0, therefore the \ system has a unique critical point at the origin (0,0). Let \[Lambda] and \ \[Mu] be the two eigenvalues of the matrix A. We distinguish the following \ three cases:", FontColor->GrayLevel[0]]], "Subsection", CellChangeTimes->{{3.654841212828898*^9, 3.654841234958022*^9}, { 3.654841581663335*^9, 3.654841585844644*^9}, {3.654841636350878*^9, 3.654841698340433*^9}, {3.699686263938445*^9, 3.699686295827509*^9}, { 3.699689214224175*^9, 3.699689255695407*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Case 1: The eigenvalues \[Lambda] and \[Mu] are real and different.", \ "Section", CellChangeTimes->{{3.654841838310152*^9, 3.654841874067608*^9}}], Cell[TextData[{ "The general solution of the system of linear differential equations is \n", Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ { RowBox[{"x", "(", "t", ")"}]}, { RowBox[{"y", "(", "t", ")"}]} }], ")"}], TraditionalForm]], FontColor->GrayLevel[0], Background->GrayLevel[0.85]], StyleBox[" = ", FontSize->24, FontColor->GrayLevel[0], Background->GrayLevel[0.85]], Cell[BoxData[ FormBox[ SubscriptBox["c", "1"], TraditionalForm]], FontColor->GrayLevel[0], Background->GrayLevel[0.85]], StyleBox[" ", FontSize->24, FontColor->GrayLevel[0], Background->GrayLevel[0.85]], Cell[BoxData[ FormBox[ SuperscriptBox["e", StyleBox["\[Lambda]t", FontSize->18]], TraditionalForm]], FontColor->GrayLevel[0], Background->GrayLevel[0.85]], StyleBox["v + ", FontSize->24, FontColor->GrayLevel[0], Background->GrayLevel[0.85]], Cell[BoxData[ FormBox[ SubscriptBox["c", "2"], TraditionalForm]], FontColor->GrayLevel[0], Background->GrayLevel[0.85]], StyleBox[" ", FontSize->24, FontColor->GrayLevel[0], Background->GrayLevel[0.85]], Cell[BoxData[ FormBox[ SuperscriptBox["e", StyleBox["\[Mu]t", FontSize->16]], TraditionalForm]], FontColor->GrayLevel[0], Background->GrayLevel[0.85]], StyleBox["w, ", FontSize->24, FontColor->GrayLevel[0], Background->GrayLevel[0.85]], StyleBox["\n\nwhere ", FontColor->GrayLevel[0], Background->GrayLevel[0.85]], Cell[BoxData[ FormBox[ SubscriptBox["c", "1"], TraditionalForm]], Background->GrayLevel[0.85]], StyleBox[" and ", FontColor->GrayLevel[0], Background->GrayLevel[0.85]], Cell[BoxData[ FormBox[ SubscriptBox["c", "2"], TraditionalForm]], Background->GrayLevel[0.85]], StyleBox[" are constant parameters, and v and w are eigenvectors \ corresponding to the eigenvalues \[Lambda] and \[Mu].", FontColor->GrayLevel[0], Background->GrayLevel[0.85]] }], "Subsection", CellChangeTimes->{{3.654842184877042*^9, 3.654842218162623*^9}, 3.6548422550517673`*^9, {3.654842314274214*^9, 3.654842418140976*^9}, 3.654845524417574*^9, 3.6548455917259912`*^9, {3.69968609863216*^9, 3.6996861053274927`*^9}}], Cell["\<\ The stability type of the critical point (0, 0) is dictated by the sign of \ the eigenvalues \[Lambda] and \[Mu].\ \>", "Subsection", CellChangeTimes->{{3.699687055414659*^9, 3.699687102301167*^9}}], Cell[TextData[{ StyleBox["a) The eigenvalues \[Lambda] and \[Mu] are both negative.\nThe \ origin is called a sink (attracting point, nodal sink, asymptotically stable \ node). ", FontColor->GrayLevel[0]], "All trajectories converge to 0 asymptotically when \ t\[RightArrow]\[Infinity]." }], "Subsection", CellChangeTimes->{{3.654841838310152*^9, 3.6548419145991793`*^9}, { 3.6548420856235533`*^9, 3.654842114540267*^9}, {3.654842446929851*^9, 3.654842470709713*^9}, {3.6548425687262163`*^9, 3.654842614807925*^9}, { 3.654842669202003*^9, 3.654842714785314*^9}, {3.654843759814921*^9, 3.654843787387291*^9}, {3.6996864155777283`*^9, 3.6996864187756367`*^9}, { 3.699686755310397*^9, 3.699686780341015*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "Example: Consider the system ", Cell[BoxData[ FormBox[ TagBox[GridBox[{ {"\[Piecewise]", GridBox[{ { RowBox[{ RowBox[{"x", "'"}], "=", RowBox[{ RowBox[{ RowBox[{"-", "3"}], "x"}], "+", RowBox[{ SqrtBox["2"], "y"}]}]}], RowBox[{"(", "1", ")"}]}, { RowBox[{ RowBox[{"y", "'"}], "=", RowBox[{ RowBox[{ SqrtBox["2"], "x"}], "-", RowBox[{"2", "y"}]}]}], RowBox[{"(", "2", ")"}]} }, AllowedDimensions->{2, Automatic}, Editable->True, 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